Are there more than 2 digits that occur infinitely often in the decimal expansion of $\sqrt{2}$?

Solution 1:

This problem is wide open. It is conjectured that every irrational algebraic number is absolutely normal (i.e. in every base, digits appear asymptoticaly with the same density). However, it is not even known whether there is any algebraic irrational with some three digits appearing infinitely many times in any base! Hence, to the best of our knowledge, every irrational algebraic number could eventually have only zeroes and ones in every base.

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